### Mandel: software for real and complex dynamics

**Mandel**is an interactive program for drawing the Mandelbrot set and Julia sets, and for illustrating and researching their mathematical properties. It is available on Linux, Windows, and Mac with a graphical user interface based on the c++ toolkit Qt by Qt Group. Features include:

- Draw the Mandelbrot set and Julia sets in two windows at the same time. Available methods of drawing include escape time, hyperbolic components or attracting basins, distance estimate, marking external rays or puzzle-pieces. Display Julia sets on the Riemann sphere.
- Trace equipotential lines and external rays, in the parameter plane and in the dynamic plane as well.
- Discuss the combinatorics of external angles, kneading sequences, and internal addresses. Illustrate the Spider Algorithm and Thurston Algorithm.
- Find centers and Misiurewicz points in the parameter plane, or periodic and preperiodic points in the dynamic plane.
- Illustrate renormalization by marking small Mandelbrot sets and small Julia sets. Mark embedded Julia sets by a kind of preperiodic renormalization.
- Display the asymptotic self-similarity at Misiurewicz points on multiple scales, and the local similarity to Julia sets.
- Draw the parameter plane and Julia sets for other one-parameter families, which satisfy critical relations or show a persistent Siegel disk. Available are unicritical polynomials, Branner-Fagella polynomials, various families of cubic and quartic polynomials, quadratic rational mappings, Newton's method, transcendental functions, a simulation of quasiconformal surgery, the Tricorn, Henon mappings, and the Barnsley–Bousch–Thurston IFS.
- Compute and display the core entropy and biaccessibility dimension for dyadic angles.
- Save images as PostScript *.eps, save or load a *.png image.
- Animated demos provide an extensive introduction to basic and advanced aspects of complex dynamics.
- Start Mandel from the GAP package IMG by Laurent Bartholdi. E.g., this package can compute a mating from combinatorial data.
- Future plans include the construction of Hubbard trees and incorporating real iteration.
- The program contains a few languages: English, German, Polish (by Adam Majewski), and Portuguese (by Atractor).

**Mandel 5.18**of May 28, 2023. The source code is available under the GNU General Public Licence. Compile it yourself on Linux, Windows, or Mac after installing Qt 4-6. For your convenience, an older Windows executable is offered in addition.

Download the

**source code**(0.5 MB) or the

**Windows exe**(5.7 MB).

**Note:**For polynomial spiders and twists, you need an old version in addition.

There is an outdated executable for

**Mac**as well (7.6 MB, Mandel 5.10, no matings).

See

**Juliette 2.0**here, a JavaScript web application providing external rays, renormalization, asymptotic and local similarity. The Java applet Juliette 1.0 is no longer supported.

**Other sites**with programs on complex dynamics or fractals:

Arnaud Chéritat, Hiroyuki Inou, Chris King, Curt McMullen, Mitsuhiro Shishikura, Xaos, Fractint.

Just for fun ... a

**PostScript program**for

*computing*M. You may change parameters by editing the source code. Download mc.eps (1.4 kb).

The DOS-program

**logistic.exe**deals with the real iteration of z

^{2}+ c, which is conjugate to the standard logistic map. It provides features like drawing q-curves, finding hyperbolic intervals or periodic points, cobweb graphics.

Download logistic.exe 2.0 of July 30, 1998. Preliminary manual logistic.ps.